Pope Francis, on July 16, 2021, issued Moto Proprio traditionis custodes, (Custodian of Tradition) and in it he abrogates "previous norms, instructions, permissions, and customs that do not conform to the provisions of the present Moto Proprio." So, does that mean that the extraordinary rite is abrogated? No, it does not! The current Moto Proprio allows for the extraordinary rite BUT - newly ordained priests who have been ordained after this date should (not must) submit a request to the local ordinary (diocesan bishop) who shall consult the Apostolic See (the Pope) before granting this authorization (Art. 4). Priests who already celebrate according to the 1962 missal again "should" (not must) "request from the diocesan bishop the authorization to continue to enjoy this faculty."
What Does This Mean to SSPX?
In reality, not much. One of the things SSPX feared was precisely what Pope Francis has just done and thus "normalization" has not yet happened. Pope Francis is, most certainly, attempting to dissuade and end "the Mass for all time" as it was called by Pope Pius V, under whom the so-called Tridentine Rite was codified. That Mass which was codified under Pope Pius V had existed since the time of the Apostles, but some variants had worked their way in through the centuries and Modernism was having its effect on the Mass - so it was codified in such a way that novel additions or changes would be prohibited. Back to the question, since SSPX has been celebrating the "extraordinary rite" since the society was canonically founded back in November of 1970 when the decree of erection of the Society was approved by Bishop Charriere of Lausanne, Geneva and Fribourg. Though this was a six year experiment, it was never abrogated and as testified by recent talks between Popes Benedict XVI and Francis, their organization is recognized - but their status remains "irregular."
What Does This Mean to FSSP?
Well, maybe nothing - maybe everything? FSSP is at the mercy of the local ordinary (diocesan bishop) and thus if said bishop decides they need to cease practicing in their diocese, FSSP is obliged to do so. To the possible advantage of FSSP, they would not enter a diocese without the bishop's explicit permission so where ever they exist, they do so under the blessing of that local bishop - unless he has a change of heart.
Time Will Tell
Only time will tell as to how much effect this latest Moto Proprio will have on the celebration of the Traditional Latin Mass.